# Vector-valued Reproducing Kernel Banach Spaces with Group Lasso Norms

## TL;DR

This paper develops a mathematical framework for vector-valued reproducing kernel Banach spaces with group lasso norms, enabling sparse multi-task learning with theoretical guarantees and new reproducing kernels.

## Contribution

It introduces RKBSs with $ ext{l}_{p,1}$-norms supporting the linear representer theorem and proposes admissible reproducing kernels for sparse multi-task learning.

## Key findings

- Established a theoretical foundation for RKBSs with group lasso norms.
- Proved the support of the linear representer theorem in this setting.
- Designed reproducing kernels suitable for sparse multi-task learning.

## Abstract

Focusing on establishing a mathematical basis for kernel methods in sparse multi-task learning, we explore the theory of vector-valued reproducing kernel Banach spaces (RKBSs) endowed with $\ell_{p,1}$-norms ($1\le p\le +\infty$), encompassing both the sparse learning case when $p=1$ and the group lasso when $p=2$. We develop RKBSs equipped with these group lasso norms that support the linear representer theorem for regularized learning frameworks. Additionally, we introduce reproducing kernels admissible for this construction. Such reproducing kernels are applicable to sparse multi-task learning with group lasso norms.